Simultaneous 4-stokes parameter determination using a single digital image

ABSTRACT

A method for determining and displaying polarization profiles of points in a scene from a single imaging detector array, which utilizes a filter system comprised of a retarder, four linear polarizers, four lenses, a color filter, camera lens and CCD video camera. Light from points in a scene are transmitted through the system and exits with attenuated intensities unique for each wavelength of the light. A narrowband color filter selects the wavelength of interest. The four lenses in the system produce four images of the scene, which are recorded as a single CCD-image. The attenuated intensities in each of the four scene-images are used to calculate the Stokes parameters for selected points in the scene for the selected wavelength. A unique pseudo-color scheme that utilizes the Poincarè sphere is used for encoding and displaying polarization parameters.

GOVERNMENT USE

The invention described here may be made, used and licensed by the orfor the U.S. Government for governmental purposes without paying us anyroyalty.

BACKGROUND

A seminal method to determine the state of polarization of a light beamusing measurable quantities is the Stokes method, which involves fourindependent intensity measurements of the light beam. Each measurementcorresponds to the intensity of the beam after it passes through each offour different filter system arrangements. The four Stokes parameters,sometimes called S₀, S₁, S₂ and S₃, are derived from these four measuredintensities and form a four-element column vector in four-dimensionalmathematical space.

Since the discovery of the Stokes method in 1852, many filter systemsbased thereon have been presented. Extracting polarization informationfrom images is not new either. However, four separate images are used tocalculate the Stokes parameters for each element in a scene. To date, amajor problem still exists in using the Stokes method for acquiringpolarization information from images. The problem occurs because ittakes time to capture separate images. In the time it takes to acquireeach image, the intensity or polarization state of points in the scenemay change. This time factor would affect polarization measurementstaken outdoors where changing sun position or cloud conditions wouldchange the intensity or polarization state of the light entering thefilter system. In the laboratory, temperature, pressure, density orconcentration variations associated with scene elements may change thepolarization state of the light entering the filter system during thetime required to record four separate images.

PRIOR ART

A relevant item of prior art is a patent to G. R. Gerhart and R. M.Matchko, “Method of Determining Polarization Profiles for PolychromaticSources. ” U.S. Pat. No. 5,734,473, issued Mar. 31, 1998.

SUMMARY

Our method and apparatus for determining and displaying polarizationprofiles of a scene from a single digital image employs a four-systemfilter-imaging array. Each of the four systems attenuates the intensityof the light transmitted through it and creates an image of the scene.The four systems operate done simultaneously in real time. Three of thefour systems consist of a linear polarizer positioned in front of animaging lens. The other system consists of a retarder and a linearpolarizer positioned in front of an imaging lens. The relative positionsof the transmission axes of the linear polarizers and the fast axis ofthe retarder determine the attenuation of the intensity of the lighttransmitted through each of the four systems. A CCD (Charged CoupledDevice) video camera, fitted with a narrow band color filter and cameralens, simultaneously captures and records the four images produced bythe four-system filter-imaging system. Each CCD video frame consists offour attenuated images of the scene. A computer program crops andregisters selected corresponding elements from each scene-image. Each ofthe four-cropped images consists of a rectangular array of pixel values(a matrix) corresponding to the attenuated intensities of the lighttransmitted through each filter. A calibration equation converts pixelvalues in each of the four matrices to optical densities and then torelative intensities. The Stokes parameters are calculated for eachpixel in the scene. Polarization parameters such as the degree ofpolarization, polarization azimuth angle and polarization ellipticityangle can be calculated for each pixel from the Stokes parametersassociated with each pixel value. A unique pseudo-color scheme thatutilizes the Poincarè sphere is used for encoding and displayingpolarization parameters in the scene. The method associates RGB valueswith the normalized values of the Stokes parameters. Our apparatus,method and polarization-encoding scheme allows one to create videoimages of changing polarization parameters in real time.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a diagram of light rays traveling from a scene through anoptical array having four optical systems, each system creating anindependent image operating according to our method.

FIG. 2 is a diagram showing light from the scene being attenuated by thefirst system, which is comprised of a linear polarizer with a horizontaltransmission axis. It also shows the image formation of the firstsystem.

FIG. 3 is a diagram showing light from the scene attenuated by thesecond system, which is comprised of a linear polarizer with a verticaltransmission axis. It also shows the image formation of the secondsystem.

FIG. 4 is a diagram showing light from the scene attenuated by thefourth system, which is comprised of a retarder whose fast axes ishorizontal and by a linear polarizer with a transmission axis at anangle θ=45 degrees relative to the horizontal direction. It also showsthe image formation of the fourth system.

FIG. 5 is a diagram showing light from the scene attenuated by the thirdsystem, which is comprised of a linear polarizer with a transmissionaxis oriented at an angle θ=45 degrees relative to the horizontaldirection. It also shows the image formation of the third system.

FIG. 6 is a diagram of the Poincarè sphere. The center of the sphere isthe origin of a rectangular Cartesian coordinate system. The sphere hasa unit radius. Every polarization state is associated with a uniquepoint in or on the sphere. The normalized Stokes parameters arerepresented by the x, y and z coordinates of a point on or inside of thesphere. Points inside the sphere correspond to partially polarizedlight, points on the surface of the sphere correspond to light that iscompletely polarized.

FIG. 7 shows a pseudo-color visualization of the surface of the Poincarèsphere when viewing the sphere along the +z axis.

FIG. 8 shows a pseudo-color visualization of the surface of the Poincarèsphere when viewing the sphere along the −z axis.

FIG. 9 shows a pseudo-color visualization of the surface and partialinterior of the Poincarè sphere when viewing the sphere along the +xaxis.

FIG. 10 shows the relationship between the polarization forms, thecorresponding pseudo-colors and the corresponding location on thePoincarè sphere when viewing the sphere along the +x axis.

FIG. 11 shows a pseudo-color view of the surface and partial interior ofthe Poincarè sphere when viewing the sphere along the −x axis.

FIG. 12 shows the relationship between the polarization forms, thecorresponding pseudo-colors and the corresponding location on thePoincarè sphere when viewing the sphere along the −x axis.

FIG. 13 shows the pseudo-coloring of the horizontal, equatorial plane ofthe Poincarè sphere. The colors along the perimeter of this plane areused to encode azimuth and ellipticity polarization angles in a scene.

DETAILED DESCRIPTION

It will be noted that FIGS. 7 through 13 depict various views of ourpseudo-color version of a Poincarè sphere. These figures, due to U.S.Patent Office regulations, must be in black and white. However, it willbe understood that our Poincarè sphere contains the colors red, blue,green and blends of these colors. In FIGS. 7 through 13, zones of ourversion of the Poincarè sphere that are primarily red, blue or green aredesignated by reference letters R, G and B , respectively.

FIG. 2 shows light that passes through the x₁-y₁ plane, such as ray 20,and is transmitted through and attenuated by linear polarizer 4 that hasits transmission axis 21 oriented at an angle θ with respect to thex₁-axis and the x₁-z₁ plane, θ being 0° in FIG. 2, such that axis 21lies along axis x₁′. The exiting attenuated light, such as ray 22, isthen transmitted through imaging lens 5, which forms attenuated image 6of scene 1. FIG. 3 shows light that passes through the x₂-y₂ plane, suchas ray 24, and is transmitted through and attenuated by linear polarizer7 that has its transmission axis 25 oriented at an angle θ with respectto the x₂-axis and the x₂-z₂ plane, θ being 90° in FIG. 3, such thataxis 25 lies along axis y₂. The exiting attenuated light, such as ray26, is then transmitted through imaging lens 8, which forms attenuatedimage 9 of scene 1.

FIG. 5 shows light that passes through the x₃-y₃ plane, such as ray 28,and is transmitted through and attenuated by linear polarizer 10 thathas its transmission axis 29 oriented at an angle θ with respect to thex₃-axis and the x₃-z₃ plane, θ being 45° in FIG. 5, where axis x₃′ isparallel to axis x₃. The exiting attenuated light, such as ray 30, isthen transmitted through imaging lens 11, which forms attenuated image12 of scene 1.

FIG. 4 shows light that passes through the x₄-y₄ plane, such as ray 32,and is transmitted through and attenuated by retarder 13 that has itsfast axis 33 oriented at an angle Ω with respect to the x₄-axis and thex₄-z₄ plane, Ω being 0° in FIG. 4, such that axis 33 lies along axisx₄′. Retarder 13 causes a phase difference ε between components of anygiven light wave passing through the system, ε having a different valuefor different wavelengths. The retarder may be of any anisotropicmaterial. Specifically, the following relationship exists for aquarter-wave plate made of quartz:

$\begin{matrix}{ɛ = {\frac{\pi}{2}\left( \frac{\lambda_{T} - 50.876}{\lambda - 50.876} \right)}} & (1)\end{matrix}$where λ is any visible wavelength and λ_(T) is that wavelength whichproduces ε=π/2, sometimes called the tuned wavelength. This relation isfurther discussed in U.S. Pat. No. 5,734,473 noted above. The exitingattenuated light, such as ray 34, is then transmitted through andattenuated by linear polarizer 14 that has its transmission axis 35oriented at an angle θ with respect to the x₄-axis and the x₄-z₄ plane,θ being 45° in FIG. 4, where axis x₄″ is parallel to axis x₄. Theexiting attenuated light, such as ray 36, is then transmitted throughimaging lens 15, which forms attenuated image 16 of scene 1.

Light from images 6, 9, 12 and 16, such as rays 23, 27, 31 and 37 (FIGS.2, 3, 5 and 4, respectively) are transmitted through a color filter 17(FIG. 1), which selects a given bandwidth, the average of which becomesλ in equation (1) above. The exiting light from color filter 17 istransmitted through a camera lens 18, which forms a collective image 38of the scene images 6, 9, 12 and 16 on the CCD array 19.

Image 38 is downloaded into a computer and a computer program cropsselected corresponding elements from each of the four scene images.Scene image 6 is cropped to form image 39, scene image 9 is cropped toform image 40, scene image 12 is cropped to form image 41 and sceneimage 16 is cropped to form image 42. The pixel values of image 39 formthe matrix M₁, the pixel values of image 40 form the matrix M₂, thepixel values of image 41 form the matrix M₃ and the pixel values ofimage 41 form the matrix M₄.

Since the Stokes parameters require intensity (I) measurements and theCCD array records RGB (red, blue and green) pixel values (X), arelationship between X and I must be obtained for the CCD array. Onecalibration method of obtaining this relationship is to pass an incidentbeam of collimated light of known intensity through neutral densityfilters of different known optical densities (Y) and record the averageX for each Y. Alternatively, instead of using an incident beam of knownintensity, one may measure the intensity of the beam exiting the neutraldensity filter. Curve-fitting yields Y as a function of X,Y=f(X).  (2)Since some CCD detectors are multi-channel arrays, a relationshipbetween X and Y must be obtained for each channel.Optical density is related to intensity through the equationI=10^(−Y)  (3)Substituting equation (2) into equation (3) yields the CCD calibrationequationI=10^(−f(X))  (4)Using equation (4), each pixel value, X, in each of the matrices M₁, M₂,M₃ and M₄ can be converted to an intensity value producing the newmatrices I₁, I₂, I₃ and I₄ respectively.

The four Stokes parameters, S₀, S₁, S₂ and S₃, are then derived from theelements of the four intensity matrices I₁, I₂, I₃ and I₄ as follows:

$\begin{matrix}{{S_{0} = {I_{1} + I_{2}}}{S_{1} = {I_{1} - I_{2}}}{S_{2} = {{2I_{3}} - S_{0}}}{S_{3} = \frac{{2I_{4}} - S_{0} - {S_{2}\cos\; ɛ}}{\sin\; ɛ}}} & (5)\end{matrix}$Each of the elements in the matrices S₀, S₁, S₂ and S₃ correspond to aparticular point in scene 1. For example, corresponding elements s⁽⁰⁾₁₁, s⁽¹⁾ ₁₁, s⁽²⁾ ₁₁ and s⁽³⁾ ₁₁ from the four Stokes parameter matricesare associated with a point (x,y) in scene 1. Therefore, thepolarization state of any point (x,y) in scene 1 can be determined from

$\begin{matrix}{{\sin\; 2\chi} = {{\frac{S_{3}}{\sqrt{S_{1}^{2} + S_{2}^{2} + S_{3}^{2}}}\mspace{14mu}\tan\; 2\psi} = {{\frac{S_{2}}{S_{1}}\mspace{14mu} P} = \frac{\sqrt{S_{1}^{2} + S_{2}^{2} + S_{3}^{2}}}{S_{0}}}}} & (6)\end{matrix}$where χ is the polarization ellipticity angle, ψ is the polarizationazimuth angle and P is the degree of polarization.

In addition to the above technique, we have invented a unique schemethat utilizes the Poincarè sphere (FIG. 6) for encoding and displayingpolarization parameters in a scene. In this scheme, the normalizedvalues of the Stokes parameters are obtained by dividing S₀, S₁, S₂ andS₃ by S₀. As shown in FIG. 6, the parameters S₁, S₂ and S₃ correspond tothe x, y and z coordinates of points inside or on of the surface of thesphere respectively. Points inside the sphere correspond to partiallypolarized light (0<P<1), whereas points on the surface of the spherecorrespond to light that is totally polarized with P=1.

The Stokes parameters are then encoded in a scene by assigning RGB (red,blue and green) values to the normalized values of S₁, S₂ and S₃ at eachpixel site in the scene as follows:R=int[127.5(1−S ₁)], G=int[127.5 (1−S ₂)] and B=int[127.5 (1−S ₃)]  (7)Where “int” is the integer function. Converting each pixel of a scene inaccordance with equation 7 will result in a color map of the scene whichcharacterizes the polarization of any selected area therein. Forexample, a given area A of the scene may have unpolarized light, whereS₁=S₂=S₃=0. Unpolarized light corresponds to middle gray (R=G=B=127) atthe center of the Poincarè sphere, and thus area A will be a middle greycolor on the aforementioned color map. Likewise, in general, anyunpolarized or weakly polarized light is middle gray or unsaturated inthe primary colors.

A method of encoding only P, the degree of polarization, is to coverteach pixel of a scene into a corresponding 8-bit digital representationby the equationpixel value=255 PEncoding the pixels in this manner will produce a monochrome orgrey-scale image, wherein the black areas correspond to light that haszero polarization, the white areas correspond to light that is 100percent polarized, and areas of varying shades of grey correspond tolight having varying degrees of polarization.

Still other options in our scheme assign RGB values to the azimuthpolarization angle or assign RGB values to the ellipticity polarizationangle. Both of these angles are essential parameters when desiring torepresent a complete polarization profile. A method of displaying eitherone these angles for each pixel in a scene is to assign a differentcolor to each specific size of that angle. The polarization azimuthangle, ψ, assumes values from 0 to 180 degrees while the ellipticityangle, χ, varies from −45 to 45 degrees. The ellipticity angle ispositive for right-handed polarization and negative for left-handedpolarization. FIG. 13 shows the pseudo-coloring of the horizontal,equatorial plane of the Poincarè sphere. We use the colors along theperimeter of this plane to encode azimuth and ellipticity polarizationangles in a scene. Each color along the perimeter of this cross-sectionof the Poincarè sphere corresponds to a unique combination of a ψ-valueand a χ-value.

The spherical polar coordinates for any point on or inside the Poincarèsphere is given byS ₁ =P cos 2 χ cos 2ξ S ₂ =P sin 2 χ cos 2ξ S ₃ =P sin 2ξ  (9)where x=S₁, y=S₂, z=S₃, P (the degree of polarization) is the radius ofthe sphere and the origin of a Cartesian coordinate system is at thecenter of the sphere. For points along the perimeter of the equatorialplane of the Poincarè sphere P=1 and χ=0. Using P=1 and χ=0 andsubstituting equation (9) into equation (7) yieldsR=int[127.5 (1−cos 2ψ)], G=int[127.5 (1−sin 2ψ)] and B=127  (10)Equation (10) contains the RGB values used to encode the ψ-values into ascene. Substituting χ for ψ in equation (10) producesR=int[127.5 (1−cos 2χ)], G=int[127.5 (1−sin 2χ)] and B=127  (11)Equation (11) contains the RGB values used to encode the χ-values into ascene.

We do not desire to be limited to the exact details of construction ormethod shown herein since obvious modifications will occur to thoseskilled in the relevant arts without departing from the spirit and scopeof the following claims.

1. A method for determining and representing in real time thepolarization profile of a scene, comprising: providing four systemshaving light attenuating optical elements; simultaneously sending raysof light incoming from the scene through all the systems, therebycreating four sets of attenuated rays; passing the sets of attenuatedrays through a color filter; passing the sets of attenuated rays to acamera CCD array which forms an image from each of the sets of rays,each of the images having a matrix of pixel values, wherein a givenpixel value in one of the images has corresponding pixel values in allothers of the images and corresponding sets of pixel values relate tosame points in the scene; calibrating a mathematical relation betweenthe pixel values and intensity values; converting the matrices of thepixel values to matrices of the intensity values; using the matrices ofintensity values to derive matrices of Stokes parameters S₀, S₁, S₂, andS₃; converting the parameter values S₁, S₂, and S₃ into values forcolors unique for each of the parameters by the formulaC=int[127.5(1−S)] where C is a color and S is S₁, S₂, or S₃; using theunique values to create a color map of the scene which represents thepolarization characteristics of the scene.
 2. The method of claim 1wherein: the four systems are a first, second, third and fourth opticalelement set; the first system comprises a first linear polarizer whosetransmission axis is oriented at an angle θ₁ with an x₁-z₁ plane alongwhich the incoming light passes; the second system comprises a secondlinear polarizer whose transmission axis is oriented at an angle θ₂ withan x₂-z₂ plane along which the incoming light passes; the third systemcomprises a third linear polarizer whose transmission axis is orientedat an angle θ₃ with an x₃-z₃ plane along which the incoming lightpasses; the fourth system comprises a retarder whose fast axis isoriented at an angle Ω with an x₄-z₄ plane along which the incominglight passes; the fourth system further comprising a fourth linearpolarizer whose transmission axis is oriented at an angle θ₄ with thex₄-z₄ plane.
 3. The method of claim 1 wherein the step of calibratingthe mathematical relation comprises: passing a beam of collimated lightthrough neutral density filters of different known optical densities andrecording the pixel value for each known intensity value for thecollimated light; and curve fitting to yield the optical density, Y, asa function of the pixel value, X, to obtain a function f(X); andsubstituting f(X) for Y in the equation I=10^(−Y), where I is theintensity.
 4. The method of claim 1 wherein the step of converting theparameter values S₁, S₂, and S₃ utilizes the colors red, green and bluefor the value C and the parameters are each associated with one of thecolors.